In the above expressions, a is called the dividend, b is called the divisor, and c is called the quotient; in the expression or a is also called the numerator and b is also called the denominator.

Conceptually, division of integers can be viewed in either of two distinct but related ways quotition and partition:

Partitioning involves taking a set of size a and forming b groups that are equal in size. The size of each group formed, c, is the quotient of a and b.

Quotition, or quotative division (also sometimes spelled quotitive) involves taking a set of size a and forming groups of size b. The number of groups of this size that can be formed, c, is the quotient of a and b.[1] (Both divisions give the same result because multiplication is commutative.)

Division is often shown in algebra and science by placing the dividend over the divisor with a horizontal line, also called a fraction bar, between them. For example, a divided by b is written

This can be read out loud as "a divided by b", "a by b" or "a over b". A way to express division all on one line is to write the dividend (or numerator), then a slash, then the divisor (or denominator), like this:

A typographical variation halfway between these two forms uses a solidus (fraction slash) but elevates the dividend, and lowers the divisor:

a⁄b

Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (typically called the numerator and denominator), and there is no implication that the division must be evaluated further. A second way to show division is to use the obelus (or division sign), common in arithmetic, in this manner:

This form is infrequent except in elementary arithmetic. ISO 80000-2-9.6 states it should not be used. The obelus is also used alone to represent the division operation itself, as for instance as a label on a key of a calculator.

In some non-English-speaking cultures, "a divided by b" is written a : b. This notation was introduced in 1631 by William Oughtred in his Clavis Mathematicae and later popularized by Gottfried Wilhelm Leibniz.[2] However, in English usage the colon is restricted to expressing the related concept of ratios (then "a is to b").

In elementary classes of some countries, the notation or is used to denote a divided by b, especially when discussing long division.[citation needed] This notation was first introduced by Michael Stifel in Arithmetica integra, published in 1544.[2]

Division is often introduced through the notion of "sharing out" a set of objects, for example a pile of sweets, into a number of equal portions. Distributing the objects several at a time in each round of sharing to each portion leads to the idea of "chunking", i.e., division by repeated subtraction.

More systematic and more efficient (but also more formalised and more rule-based, and more removed from an overall holistic picture of what division is achieving), a person who knows the multiplication tables can divide two integers using pencil and paper using the method of short division, if the divisor is simple. Long division is used for larger integer divisors. If the dividend has a fractional part (expressed as a decimal fraction), one can continue the algorithm past the ones place as far as desired. If the divisor has a fractional part, we can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction.

A person can calculate division with an abacus by repeatedly placing the dividend on the abacus, and then subtracting the divisor the offset of each digit in the result, counting the number of divisions possible at each offset.

A person can use logarithm tables to divide two numbers, by subtracting the two numbers' logarithms, then looking up the antilogarithm of the result.

A person can calculate division with a slide rule by aligning the divisor on the C scale with the dividend on the D scale. The quotient can be found on the D scale where it is aligned with the left index on the C scale. The user is responsible, however, for mentally keeping track of the decimal point.

Modern computers compute division by methods that are faster than long division: see Division algorithm.

In modular arithmetic, some numbers have a multiplicative inverse with respect to the modulus. We can calculate division by multiplication in such a case. This approach is useful in computers that do not have a fast division instruction.

The Euclidean division is the mathematical formulation of the outcome of the usual process of division of integers. It asserts that, given two integers, a, the dividend, and b, the divisor, such that b ≠ 0, there are unique integers q, the quotient, and r, the remainder, such that a = bq + r and 0 ≤ r < | b |, where | b | denotes the absolute value of b.

Division of integers is not closed. Apart from division by zero being undefined, the quotient is not an integer unless the dividend is an integer multiple of the divisor. For example 26 cannot be divided by 11 to give an integer. Such a case uses one of five approaches:

Give the answer as a fraction representing a rational number, so the result of the division of 26 by 11 is But, usually, the resulting fraction should be simplified: the result of the division of 52 by 22 is also . This simplification may be done by factoring out the greatest common divisor.

Give the integer quotient as the answer, so This is sometimes called integer division.

Dividing integers in a computer program requires special care. Some programming languages, such as C, treat integer division as in case 5 above, so the answer is an integer. Other languages, such as MATLAB and every computer algebra system return a rational number as the answer, as in case 3 above. These languages also provide functions to get the results of the other cases, either directly or from the result of case 3.

Names and symbols used for integer division include div, /, \, and %. Definitions vary regarding integer division when the dividend or the divisor is negative: rounding may be toward zero (so called T-division) or toward −∞ (F-division); rarer styles can occur – see Modulo operation for the details.

Divisibility rules can sometimes be used to quickly determine whether one integer divides exactly into another.

Division of any number by zero (where the divisor is zero) is undefined. This is because zero multiplied by any finite number always results in a product of zero. Entry of such an expression into most calculators produces an error message.

One can define a division operation for matrices. The usual way to do this is to define A / B = AB−1, where B−1 denotes the inverse of B, but it is far more common to write out AB−1 explicitly to avoid confusion.

Because matrix multiplication is not commutative, one can also define a left division or so-called backslash-division as A \ B = A−1B. For this to be well defined, B−1 need not exist, however A−1 does need to exist. To avoid confusion, division as defined by A / B = AB−1 is sometimes called right division or slash-division in this context.

Note that with left and right division defined this way, A/(BC) is in general not the same as (A/B)/C and nor is (AB)\C the same as A\(B\C), but A/(BC) = (A/C)/B and (AB)\C = B\(A\C).

To avoid problems when A−1 and/or B−1 do not exist, division can also be defined as multiplication with the pseudoinverse, i.e., A / B = AB+ and A \ B = A+B, where A+ and B+ denote the pseudoinverse of A and B.

In abstract algebras such as matrix algebras and quaternion algebras, fractions such as are typically defined as or where is presumed an invertible element (i.e., there exists a multiplicative inverse such that where is the multiplicative identity). In an integral domain where such elements may not exist, division can still be performed on equations of the form or by left or right cancellation, respectively. More generally "division" in the sense of "cancellation" can be done in any ring with the aforementioned cancellation properties. If such a ring is finite, then by an application of the pigeonhole principle, every nonzero element of the ring is invertible, so division by any nonzero element is possible in such a ring. To learn about when algebras (in the technical sense) have a division operation, refer to the page on division algebras. In particular Bott periodicity can be used to show that any realnormed division algebra must be isomorphic to either the real numbers R, the complex numbersC, the quaternionsH, or the octonionsO.